Gravity Analysis Regge Exact Midpoint M2 Ttidentity4 D Exact Midpoint Bloch M2 E

The midpoint identity

In the Recognition Science framework's machine-checked library of formal theorems, a key result simplifies a complicated expression used in the analysis of gravitational waves. The theorem, named exactMidpointBlochM2_eq_closedForm_of_symmetric, states that a particular quantity, the exact midpoint Bloch m² term, can be replaced by a much simpler closed-form expression. This simplification is valid only when the input matrix H, which represents a metric-like perturbation, is symmetric, meaning it is unchanged when its rows and columns are swapped.

The exact midpoint Bloch m² term is a sum over many combinations of indices, arising from a discrete model of spacetime. The closed form, by contrast, is a compact expression built from a few standard matrix operations: the squared Frobenius norm (sum of squares of all entries), the squared norm of a wave vector, a quadratic form, and the trace. The theorem proves these two very different-looking formulas are exactly equal for any symmetric matrix H and any wave vector k.

This identity is not an approximation. It is a proven equality, derived within the framework's formal system. The proof relies on the symmetry of H, which allows a reorganization of the summation terms. The theorem is a key step in a larger effort to show that certain complicated gravitational expressions reduce to simple, interpretable forms, such as a direct product of the matrix size and the wave's magnitude.

What the theorem does not claim is just as important. It does not say the closed form is valid for non-symmetric matrices; the symmetry condition is essential. It does not assert any physical law about gravity itself, only an algebraic identity between two defined quantities. It also does not claim that the closed form is the only possible simplification, or that the exact midpoint term has a unique physical meaning beyond its definition.

theorem exactMidpointBlochM2_eq_closedForm_of_symmetric
    (H : Mat4) (k : Wave4) (hsym : IsSymmetric H) :
    exactMidpointBlochM2 H k = closedForm H k := by
  calc
    exactMidpointBlochM2 H k = biquad m2Coeff H k :=
      exactMidpointBlochM2_eq_biquad H k
    _ = biquad explicitM2Coeff H k :=
      biquad_congr m2Coeff_eq_explicitM2Coeff H k
    _ = biquad (symFull explicitM2Coeff) H k :=
      (biquad_symFull explicitM2Coeff H k hsym).symm
    _ = biquad (symFull closedCoeff) H k :=
      biquad_congr symFull_explicit_eq_symFull_closed H k
    _ = biquad closedCoeff H k := biquad_symFull closedCoeff H k hsym
    _ = closedForm H k := biquad_closedCoeff_eq_closedForm H k

theorem exactMidpointBlochM2_eq_closedForm_of_symmetric
    (H : Mat4) (k : Wave4) (hsym : IsSymmetric H) :
    exactMidpointBlochM2 H k = closedForm H k := by
  calc
    exactMidpointBlochM2 H k = biquad m2Coeff H k :=
      exactMidpointBlochM2_eq_biquad H k
    _ = biquad explicitM2Coeff H k :=
      biquad_congr m2Coeff_eq_explicitM2Coeff H k
    _ = biquad (symFull explicitM2Coeff) H k :=
      (biquad_symFull explicitM2Coeff H k hsym).symm
    _ = biquad (symFull closedCoeff) H k :=
      biquad_congr symFull_explicit_eq_symFull_closed H k
    _ = biquad closedCoeff H k := biquad_symFull closedCoeff H k hsym
    _ = closedForm H k := biquad_closedCoeff_eq_closedForm H k

What this page does not claim

The theorem does not apply to non-symmetric matrices. The theorem does not assert any physical law about gravity. The theorem does not claim the closed form is the only possible simplification.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/Analysis/ReggeExactMidpointM2TTIdentity4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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